Determine if the following sequences are monotonic and/or bounded.
(a) {-n2}∞n=0
(b) {( -1)n+1}∞n=1
(c) {2/n2}∞n=5
Solution
{-n2}∞n=0
This sequence is a decreasing sequence (and therefore monotonic) because,
-n2 > - (n +1)2
for each n.
As well, since the sequence terms will be either zero or negative this type of sequence is bounded above. We can make use of any positive number or zero as the bound, M, though, it's standard to choose the smallest possible bound if we can and it is a nice number. Thus, we'll choose M = 0 since,
-n2 ≤ 0 for every n
This type of sequence is not bounded below though as we can always get below any potential bound by taking n large enough. Hence, when the sequence is bounded above it is not bounded.
As a side note we can as well note that this sequence diverges (to -∞ if we want to be specific).
(b) {( -1)n+1}∞n=1
The sequence terms in this type of sequence alternate in between 1 and -1 and thus the sequence is neither a decreasing sequence nor increasing sequence. As the sequence is neither an increasing nor decreasing sequence it is not called as a monotonic sequence.
Though, the sequence is bounded since it is bounded above by 1 and bounded below by -1.
Once again, we can note that this sequence is as well divergent.
(c) {2/n2}∞n=5
The above sequence is a decreasing sequence and therefore monotonic since,
(2 / n2) > (2 / (n+1)2)
The terms in this sequence are all positive and thus it is bounded below by zero. As well, since the sequence is a decreasing sequence the first sequence term will be the largest and thus we can see that the sequence will as well be bounded above by 2/25. Hence, this sequence is bounded.
We can as well take a quick limit and note that this sequence converges and the limit of it is zero.